MEAN-VARIANCE PORTFOLIO SELECTION: DE FINETTI SCOOPS MARKOWITZ

1
MEAN-VARIANCE PORTFOLIO SELECTIONDE FINETTI
SCOOPS MARKOWITZ

• November 2006

2
OUTLINE

• Introduction
• Historical Note
• Geometric View
• Uncorrelated Assets
• Two Assets / Three Assets / Four Assets
• Correlated Assets
• Two Assets / Three Assets
• Last Segment
• Markowitzs 2D Analysis
• Internal Solutions
• Edge Solutions
• Pressaccos 3D Extension
• Conclusions

3
Introduction
Bruno de Finetti is generally regarded as the
finest Italian mathematician of the 20th
century Mark Rubinstein

• Acknowledgments by Mark Rubinstein
• early work on martingales (1939)
• mean-variance portfolio theory (1940)
• portfolio variance as a sum of covariances
• concept of mean-variance efficiency
• normality of returns
• implications of fat tails
• bounds on negative correlation coefficients
• early version of the critical line algorithm
• notion of absolute risk aversion (1952)
• early work on optimal dividend policy (1957)
• early work on Samuelsons consumption loan model
  of interest rates (1956)
• Acknowledgments by Harry Markowitz
• mean-variance portfolio analysis using correlated
  risks
• efficient frontier for the case with uncorrelated
  risks
• special case of the Kuhn-Tucker theorem
• Critique by Harry Markowitz
• the critical lines last segment does not
  necessarily lies inside the legitimate set

4
Introduction

• Other points in the 1940 article on The Problem
  of Full-Risk Insurances (Il problema dei pieni).
• Bruno de Finetti
• anticipated two classical steps in capital
  allocation. What he called the problem of
  relative full-risk insurances is the problem of
  finding the efficient frontier, while what he
  called the problem of absolute full-risk
  insurances is that of choosing the optimal
  portfolio along the efficient frontier
• defined and analyzed the variable t (called the
  safety degree) which corresponds to the Sharpe
  ratio
• sketched what can be considered an embryo of Nash
  equilibrium theory
• If we consider several insurers who offer
  reinsurance to one another and everyone of these
  behaves when determining the levels of
  full-risk insurances in the optimal way
  suggested by our preceding results, that is each
  one seeks the solution which is the most
  advantageous for his own ends, it cannot taken
  for granted that they will actually succeed in
  reaching their aim This statement is related, as
  a particular case, to my preceding considerations
  intended to disprove the fundamental principle of
  a liberal economy, according to which the free
  play of single egoisms would produce a collective
  optimum .... Therefore, we will now try to see
  whether addressing the problem simultaneously,
  rather than unilaterally, and using the
  conclusions of such a study to formulate a mutual
  convention rather than many unilateral decisions
  it is possible to establish a different
  criterion which, for all and everyone, is more
  advantageous than the one we found before.
• In this presentation, I will only use charts to
  explain de Finettis paper and Markowitzs
  critique.
• The analytical framework is reviewed in my paper
  on Bruno de Finetti and the Case of the Critical
  Lines Last Segment

5
Historical Note

• It is not easy to find references to de Finettis
  1940 paper
• An exception is given by a short notice in the
  Journal of the Institute of Actuaries (1947)
• An important acknowledgement comes from Hans
  Bühlmann, who dealt with the retention problem in
  his 1970 book
• In Chapter 5, Retention and Reserves, Bühlmann
  ... gives a method of de Finetti from a 1940
  Italian paper. Although these ideas appear to be
  of great use and are known widely in Europe, we
  are indebted to Bühlmann for making them
  available in English.
• De Finettis contribution to portfolio theory has
  been pointed out by Flavio Pressacco (1986)
• The name of H. Markowitz (1952) is famous all
  over the world for his mean variance approach to
  the portfolio selection problem, a milestone in
  the analysis of relevant economic problems under
  uncertainty. It is perhaps surprising to find
  that more than ten years earlier B. de Finetti
  (1940) used the same approach to study a key
  problem in proportional reinsurance the optimal
  retention problem.
• The forward to a paper by Swiss Re (2003), where
  Hans Bühlmann worked, refers to de Finettis
  paper
• The publication is based on Bruno de Finettis
  work on the establishment of optimal proportional
  retentions which was published in a 1940 article
  entitled Il problema dei pieni. ... Because de
  Finettis work remained virtually unknown outside
  universities, actuaries and other professionals
  are still to a large extent unfamiliar with
  practicable rules deriving from his observations.
  For this reason, even readers with knowledge of
  risk theory will also find something new in the
  following work.

6
Geometric View

• The reinsurance problem studied by de Finetti
  (1940) is very similar to, but formally different
  from, the portfolio selection problem analyzed by
  Markowitz (1952).
• In both problems the goal is to minimize the
  variance of a portfolios return, for a given
  level of expected return, subject to some linear
  constraints (a quadratic programming problem).
• Markowitz studies how to select efficient
  portfolios by investing a unit of capital, while
  de Finetti considers a given portfolio and
  studies how to revise its weights, by selling
  (i.e. reinsuring) some of the securities (i.e.
  insurance policies), in order to obtain efficient
  portfolios.
• Therefore, both de Finetti and Markowitz study
  how to find the optimal weights of efficient
  portfolios.
• Markowitz (2006) approves the de Finetti analysis
  for the no-correlation case (For the case of
  uncorrelated risks, de Finetti solved the problem
  of computing the set of mean-variance efficient
  portfolios.)
• However, the solution of the reinsurance problem
  (i.e. the reinsurance frontier) will not
  necessarily represent the set of all the
  efficient portfolios (i.e. the efficient
  frontier) since we do not know if the initial
  portfolio is efficient or not.

7
Geometric View

• Markowitz (2006) gives the following numerical
  example, where the rates of return of two
  securities have means equal to 1 (µ1 µ2 1)
  and standard deviations equal to 1 and 2,
  respectively (s1 1, s2 2).

• The geometric solution may be better seen in
  Excel.

8
Last Segment

• Is de Finettis last segment conjecture not
  correct?

• Bruno de Finetti The geometric interpretation
  given for the no-correlation case still holds,
  since the line of optimal points of s for given
  E is a continuous broken line which links the
  starting point to the origin (0, 0, ..., 0).

• Harry Markowitz He tells where the efficient
  set starts how it traces out a sequence of
  connected straight line segments and then
  describes how it ends, i.e., the general location
  of the last segment of the path. I will refer to
  de Finettis statement on the latter matter as
  de Finettis last segment conjecture. It is
  not correct.

• A controversy based on one statement

• Bruno de Finetti until we will finally proceed
  on the last segment inside the hypercube, along
  the straight line K1 K2 ... Kn in
  Italian ... finché finalmente si percorrerà
  lultimo tratto allinterno dellipercubo lungo
  la retta K1 K2 ... Kn

• Harry Markowitz ... in the de Finetti model
  with correlated risks, unlike the case with
  uncorrelated risks, it is possible for the last
  segment to approach the zero portfolio along the
  edge of the square or along the face or edge of
  the cube or hypercube, rather than through its
  interior.

• Did de Finetti solve the reinsurance problem for
  the case of correlated assets?

• Bruno de Finetti (correlation case)the
  determination of the levels of full-risk
  insurances entails the laborious, though per se
  elementary, problem of solving a system of linear
  equations.

• Harry Markowitz De Finetti solved the problem
  of computing mean-variance efficient frontiers
  assuming uncorrelated risks. While he understood
  and explained the importance of the case with
  correlated risks, he did not provide an algorithm
  for this case.

9
Last Segment

• There is no doubt that, in his geometrical
  representation, de Finetti locates the last
  segment inside the hypercube and along the pure
  critical line.
• It could be argued that de Finettis statement
  does not implicitly exclude the possibility of
  the last segment lying on the hypercube
• the adverb inside surely rules out the points
  outside the hypercube but it does not necessarily
  exclude the points on the edges of the hypercube
• the adverb along does not rule out the
  possibility of the last segment touching the pure
  critical line at the origin, through which the
  pure critical line must always pass.
• However, the most reasonable justifications seem
  to be the following
• the cases in which all the policies need to be
  reinsured is unlikely to be important from a
  practical standpoint, even if the academic
  relevance of the last segment (which leads to the
  risk-free portfolio) is undeniable.
• The following de Finettis statement gives
  support to this interpretation
• ... all these equalities would be satisfied only
  when all the contracts needed to be reinsured, a
  point that is out of the question in practice.
• even if we were interested in the last segment
  from a practical standpoint, the circumstances
  under which de Finettis statement is not correct
  are relatively rare.

10
Conclusions

• Despite the controversy on the position of the
  last segment in the correlation case, Markowitz
  acknowledges the merits of de Finettis paper and
  titles his 2006 article as De Finetti Scoops
  Markowitz
• He recognizes that de Finetti solved the problem
  of computing mean-variance efficient reinsurance
  frontiers for the case of uncorrelated risks and
  gives him credit for outlining some of the
  properties that characterize the solution of the
  reinsurance problem with correlated risks.
• He also acknowledges that de Finetti worked out a
  special case of the Kuhn-Tucker theorem
• Finally, he points out that de Finettis
  achievements did not benefit from an environment
  in which the existence of the Kuhn-Tucker
  theorem and the success of linear programming
  encouraged a presumption that a neat quadratic
  programming algorithm existed if we persisted in
  seeking it.
• Even if he did not fully work out the critical
  line algorithm, Bruno de Finetti did actually lay
  - in 1940 - the foundations of modern finance
  theory, as acknowledged by Mark Rubinstein.
• After Arrow-Debreu, Modigliani-Miller,
  Black-Scholes, ... has the time finally come to
  talk about the de Finetti - Markowitz portfolio
  selection theory?

11
(No Transcript)